Abstract: We consider the smallest values taken by the Jones index for an inclusion oflocal conformal nets of von Neumann algebras on S^1 and show that these valuesare quite more restricted than for an arbitrary inclusion of factors. Below 4,the only non-integer admissible value is 4\cos^2 \pi-10, which is known to beattained by a certain coset model. Then no index value is possible in theinterval between 4 and 3 +\sqrt{3}. The proof of this result based on\alpha-induction arguments. In the case of values below 4 we also give a secondproof of the result. In the course of the latter proof we classify all possibleunitary braiding symmetries on the A D E tensor categories, namely the onesassociated with the even vertices of the A n, D {2n}, E 6, E 8 Dynkin diagrams.